714 lines
23 KiB
C++
714 lines
23 KiB
C++
// MFEM Example 27 - Serial Version
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//
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// Compile with: make ex27
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//
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// Sample runs: ex27
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// ex27 -dg
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// ex27 -dg -dbc 8 -nbc -2
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// ex27 -rbc-a 1 -rbc-b 8
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//
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// Description: This example code demonstrates the use of MFEM to define a
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// simple finite element discretization of the Laplace problem
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// -Delta u = 0 with a variety of boundary conditions.
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// Specifically, we discretize using a FE space of the specified
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// order using a continuous or discontinuous space. We then
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// apply Dirichlet, Neumann (both homogeneous and inhomogeneous),
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// Robin, and Periodic boundary conditions on different portions
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// of a predefined mesh.
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//
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// The predefined mesh consists of a rectangle with two
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// holes removed (see below). The narrow ends of the
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// mesh are connected to form a Periodic boundary
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// condition. The lower edge (tagged with attribute 1)
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// receives an inhomogeneous Neumann boundary condition.
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// A Robin boundary condition is applied to upper edge
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// (attribute 2). The circular hole on the left
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// (attribute 3) enforces a Dirichlet boundary
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// condition. Finally, a natural boundary condition, or
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// homogeneous Neumann BC, is applied to the circular
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// hole on the right (attribute 4).
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//
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// Attribute 3 ^ y Attribute 2
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// \ | /
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// +-----------+-----------+
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// | \_ | _ |
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// | / \ | / \ |
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// <--+---+---+---+---+---+---+--> x
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// | \_/ | \_/ |
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// | | \ |
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// +-----------+-----------+ (hole radii are
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// / | \ adjustable)
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// Attribute 1 v Attribute 4
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//
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// The boundary conditions are defined as (where u is
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// the solution field):
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// Dirichlet: u = d
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// Neumann: n.Grad(u) = g
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// Robin: n.Grad(u) + a u = b
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//
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// The user can adjust the values of 'd', 'g', 'a', and
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// 'b' with command line options.
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//
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// This example highlights the differing implementations of
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// boundary conditions with continuous and discontinuous Galerkin
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// formulations of the Laplace problem.
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//
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// We recommend viewing examples 1 and 14 before viewing this
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// example.
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#include "mfem.hpp"
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#include <fstream>
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#include <iostream>
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using namespace std;
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using namespace mfem;
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static double a_ = 0.2;
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// Normal to hole with boundary attribute 4
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void n4Vec(const Vector &x, Vector &n) { n = x; n[0] -= 0.5; n /= -n.Norml2(); }
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Mesh * GenerateSerialMesh(int ref);
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double IntegrateBC(GridFunction & x, Array<int> &bdr, bool h1);
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double IntegrateNGradBC(GridFunction &u, VectorCoefficient &nCoef,
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Array<int> &bdr, bool h1,
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OperatorHandle &M, double a = 0.0);
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int main(int argc, char *argv[])
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{
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// 1. Parse command-line options.
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int ser_ref_levels = 2;
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int order = 1;
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double sigma = -1.0;
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double kappa = -1.0;
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bool h1 = true;
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bool visualization = true;
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double mat_val = 1.0;
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double dbc_val = 0.0;
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double nbc_val = 1.0;
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double rbc_a_val = 1.0; // du/dn + a * u = b
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double rbc_b_val = 1.0;
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OptionsParser args(argc, argv);
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args.AddOption(&h1, "-h1", "--continuous", "-dg", "--discontinuous",
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"Select continuous \"H1\" or discontinuous \"DG\" basis.");
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args.AddOption(&order, "-o", "--order",
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"Finite element order (polynomial degree) or -1 for"
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" isoparametric space.");
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args.AddOption(&sigma, "-s", "--sigma",
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"One of the two DG penalty parameters, typically +1/-1."
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" See the documentation of class DGDiffusionIntegrator.");
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args.AddOption(&kappa, "-k", "--kappa",
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"One of the two DG penalty parameters, should be positive."
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" Negative values are replaced with (order+1)^2.");
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args.AddOption(&ser_ref_levels, "-rs", "--refine-serial",
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"Number of times to refine the mesh uniformly in serial.");
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args.AddOption(&mat_val, "-mat", "--material-value",
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"Constant value for material coefficient "
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"in the Laplace operator.");
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args.AddOption(&dbc_val, "-dbc", "--dirichlet-value",
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"Constant value for Dirichlet Boundary Condition.");
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args.AddOption(&nbc_val, "-nbc", "--neumann-value",
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"Constant value for Neumann Boundary Condition.");
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args.AddOption(&rbc_a_val, "-rbc-a", "--robin-a-value",
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"Constant 'a' value for Robin Boundary Condition: "
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"du/dn + a * u = b.");
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args.AddOption(&rbc_b_val, "-rbc-b", "--robin-b-value",
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"Constant 'b' value for Robin Boundary Condition: "
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"du/dn + a * u = b.");
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args.AddOption(&a_, "-a", "--radius",
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"Radius of holes in the mesh.");
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args.AddOption(&visualization, "-vis", "--visualization", "-no-vis",
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"--no-visualization",
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"Enable or disable GLVis visualization.");
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args.Parse();
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if (!args.Good())
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{
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args.PrintUsage(mfem::out);
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return 1;
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}
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if (kappa < 0 && !h1)
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{
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kappa = (order+1)*(order+1);
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}
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args.PrintOptions(mfem::out);
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if (a_ < 0.01)
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{
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mfem::out << "Hole radius too small, resetting to 0.01.\n";
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a_ = 0.01;
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}
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if (a_ > 0.49)
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{
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mfem::out << "Hole radius too large, resetting to 0.49.\n";
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a_ = 0.49;
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}
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// 2. Construct the (serial) mesh and refine it if requested.
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Mesh *mesh = GenerateSerialMesh(ser_ref_levels);
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int dim = mesh->Dimension();
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// 3. Define a finite element space on the serial mesh. Here we
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// use either continuous Lagrange finite elements or discontinuous
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// Galerkin finite elements of the specified order.
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FiniteElementCollection *fec =
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h1 ? (FiniteElementCollection*)new H1_FECollection(order, dim) :
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(FiniteElementCollection*)new DG_FECollection(order, dim);
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FiniteElementSpace fespace(mesh, fec);
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int size = fespace.GetTrueVSize();
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mfem::out << "Number of finite element unknowns: " << size << endl;
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// 4. Create "marker arrays" to define the portions of the boundary
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// associated with each type of boundary condition. These arrays
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// have an entry corresponding to each boundary attribute.
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// Placing a '1' in an entry marks that attribute as being
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// active, '0' is inactive.
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Array<int> nbc_bdr(mesh->bdr_attributes.Max());
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Array<int> rbc_bdr(mesh->bdr_attributes.Max());
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Array<int> dbc_bdr(mesh->bdr_attributes.Max());
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nbc_bdr = 0; nbc_bdr[0] = 1;
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rbc_bdr = 0; rbc_bdr[1] = 1;
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dbc_bdr = 0; dbc_bdr[2] = 1;
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Array<int> ess_tdof_list(0);
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if (h1 && mesh->bdr_attributes.Size())
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{
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// For a continuous basis the linear system must be modifed to enforce
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// an essential (Dirichlet) boundary condition. In the DG case this is
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// not necessary as the boundary condition will only be enforced weakly.
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fespace.GetEssentialTrueDofs(dbc_bdr, ess_tdof_list);
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}
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// 5. Setup the various coefficients needed for the Laplace operator and
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// the various boundary conditions. In general these coefficients could
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// be functions of position but here we use only constants.
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ConstantCoefficient matCoef(mat_val);
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ConstantCoefficient dbcCoef(dbc_val);
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ConstantCoefficient nbcCoef(nbc_val);
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ConstantCoefficient rbcACoef(rbc_a_val);
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ConstantCoefficient rbcBCoef(rbc_b_val);
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ProductCoefficient m_nbcCoef(matCoef, nbcCoef);
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ProductCoefficient m_rbcACoef(matCoef, rbcACoef);
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ProductCoefficient m_rbcBCoef(matCoef, rbcBCoef);
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// 6. Define the solution vector u as a finite element grid function
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// corresponding to fespace. Initialize u with initial guess of zero.
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GridFunction u(&fespace);
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u = 0.0;
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// 7. Set up the bilinear form a(.,.) on the finite element space
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// corresponding to the Laplacian operator -Delta, by adding the Diffusion
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// domain integrator.
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BilinearForm a(&fespace);
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a.AddDomainIntegrator(new DiffusionIntegrator(matCoef));
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if (h1)
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{
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// Add a Mass integrator on the Robin boundary
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a.AddBoundaryIntegrator(new MassIntegrator(m_rbcACoef), rbc_bdr);
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}
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else
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{
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// Add the interfacial portion of the Lapalce operator
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a.AddInteriorFaceIntegrator(new DGDiffusionIntegrator(matCoef,
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sigma, kappa));
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// Counteract the n.Grad(u) term on the Dirichlet portion of the boundary
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a.AddBdrFaceIntegrator(new DGDiffusionIntegrator(matCoef, sigma, kappa),
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dbc_bdr);
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// Augment the n.Grad(u) term with a*u on the Robin portion of boundary
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a.AddBdrFaceIntegrator(new BoundaryMassIntegrator(m_rbcACoef),
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rbc_bdr);
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}
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a.Assemble();
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// 8. Assemble the linear form for the right hand side vector.
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LinearForm b(&fespace);
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if (h1)
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{
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// Set the Dirchlet values in the solution vector
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u.ProjectBdrCoefficient(dbcCoef, dbc_bdr);
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// Add the desired value for n.Grad(u) on the Neumann boundary
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b.AddBoundaryIntegrator(new BoundaryLFIntegrator(m_nbcCoef), nbc_bdr);
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// Add the desired value for n.Grad(u) + a*u on the Robin boundary
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b.AddBoundaryIntegrator(new BoundaryLFIntegrator(m_rbcBCoef), rbc_bdr);
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}
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else
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{
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// Add the desired value for the Dirchlet boundary
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b.AddBdrFaceIntegrator(new DGDirichletLFIntegrator(dbcCoef, matCoef,
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sigma, kappa),
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dbc_bdr);
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// Add the desired value for n.Grad(u) on the Neumann boundary
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b.AddBdrFaceIntegrator(new BoundaryLFIntegrator(m_nbcCoef),
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nbc_bdr);
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// Add the desired value for n.Grad(u) + a*u on the Robin boundary
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b.AddBdrFaceIntegrator(new BoundaryLFIntegrator(m_rbcBCoef),
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rbc_bdr);
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}
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b.Assemble();
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// 9. Construct the linear system.
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OperatorPtr A;
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Vector B, X;
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a.FormLinearSystem(ess_tdof_list, u, b, A, X, B);
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#ifndef MFEM_USE_SUITESPARSE
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// 10. Define a simple symmetric Gauss-Seidel preconditioner and use it to
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// solve the system AX=B with PCG in the symmetric case, and GMRES in the
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// non-symmetric one.
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{
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GSSmoother M((SparseMatrix&)(*A));
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if (sigma == -1.0)
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{
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PCG(*A, M, B, X, 1, 500, 1e-12, 0.0);
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}
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else
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{
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GMRES(*A, M, B, X, 1, 500, 10, 1e-12, 0.0);
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}
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}
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#else
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// 11. If MFEM was compiled with SuiteSparse, use UMFPACK to solve the
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// system.
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UMFPackSolver umf_solver;
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umf_solver.Control[UMFPACK_ORDERING] = UMFPACK_ORDERING_METIS;
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umf_solver.SetOperator(*A);
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umf_solver.Mult(B, X);
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#endif
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// 12. Recover the grid function corresponding to U. This is the
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// local finite element solution.
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a.RecoverFEMSolution(X, b, u);
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// 13. Build a mass matrix to help solve for n.Grad(u) where 'n' is
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// a surface normal.
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BilinearForm m(&fespace);
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m.AddDomainIntegrator(new MassIntegrator);
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m.Assemble();
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ess_tdof_list.SetSize(0);
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OperatorPtr M;
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m.FormSystemMatrix(ess_tdof_list, M);
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// 14. Compute the various boundary integrals.
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mfem::out << endl << "Verifying boundary conditions:" << endl;
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{
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// Integrate the solution on the Dirichlet boundary and compare
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// to the expected value. The factor of (2 pi a) divides by the
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// circumference of the hole.
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double dbc_int = IntegrateBC(u, dbc_bdr, h1) / (2.0 * M_PI * a_);
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double dbc_err = fabs(dbc_int - dbc_val);
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bool hom_dbc = (dbc_val == 0.0);
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dbc_err /= hom_dbc ? 1.0 : fabs(dbc_val);
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mfem::out << "dbc " << dbc_int << ", "
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<< (hom_dbc ? "absolute" : "relative")
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<< " error " << dbc_err << endl;
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}
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{
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// Integrate n.Grad(u) with n = (0, -1) on the inhomogeneous
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// Neumann boundary and compare to the expected value.
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Vector nVec(2); nVec[0] = 0.0; nVec[1] = -1.0;
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VectorConstantCoefficient nCoef(nVec);
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// The factor of 0.5 divides by the length of the boundary.
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double nbc_int = IntegrateNGradBC(u, nCoef, nbc_bdr, h1, M) * 0.5;
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double nbc_err = fabs(nbc_int - nbc_val);
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bool hom_nbc = (nbc_val == 0.0);
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nbc_err /= hom_nbc ? 1.0 : fabs(nbc_val);
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mfem::out << "nbc " << nbc_int << ", "
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<< (hom_nbc ? "absolute" : "relative")
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<< " error " << nbc_err << endl;
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}
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{
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// Integrate n.Grad(u) with n given by the function n4Vec on the
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// homogeneous Neumann boundary and compare to the expected
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// value of zero.
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Array<int> nbc0_bdr(mesh->bdr_attributes.Max());
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nbc0_bdr = 0;
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nbc0_bdr[3] = 1;
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// The factor of (2 pi a) divides by the circumference of the hole.
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VectorFunctionCoefficient n0Coef(2, n4Vec);
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double nbc_int = IntegrateNGradBC(u, n0Coef, nbc0_bdr, h1, M) /
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(2.0 * M_PI * a_);
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double nbc_err = fabs(nbc_int);
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bool hom_nbc = true;
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mfem::out << "nbc0 " << nbc_int << ", "
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<< (hom_nbc ? "absolute" : "relative")
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<< " error " << nbc_err << endl;
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}
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{
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// Integrate n.Grad(u) + a * u with n = (0, 1) on the Robin
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// boundary and compare to the expected value.
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Vector nVec(2); nVec[0] = 0.0; nVec[1] = 1.0;
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VectorConstantCoefficient nCoef(nVec);
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// The factor of 0.5 divides by the length of the boundary.
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double rbc_int = IntegrateNGradBC(u, nCoef, rbc_bdr, h1, M, rbc_a_val)
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* 0.5;
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double rbc_err = fabs(rbc_int - rbc_b_val);
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bool hom_rbc = (rbc_b_val == 0.0);
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rbc_err /= hom_rbc ? 1.0 : fabs(rbc_b_val);
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mfem::out << "rbc " << rbc_int << ", "
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<< (hom_rbc ? "absolute" : "relative")
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<< " error " << rbc_err << endl;
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}
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// 15. Save the refined mesh and the solution in parallel. This output can
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// be viewed later using GLVis: "glvis -np <np> -m mesh -g sol".
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{
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ofstream mesh_ofs("refined.mesh");
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mesh_ofs.precision(8);
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mesh->Print(mesh_ofs);
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ofstream sol_ofs("sol.gf");
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sol_ofs.precision(8);
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u.Save(sol_ofs);
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}
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// 16. Send the solution by socket to a GLVis server.
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if (visualization)
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{
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string title_str = h1 ? "H1" : "DG";
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char vishost[] = "localhost";
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int visport = 19916;
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socketstream sol_sock(vishost, visport);
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sol_sock.precision(8);
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sol_sock << "solution\n" << *mesh << u
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<< "window_title '" << title_str << " Solution'"
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<< " keys 'mmc'" << flush;
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}
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// 17. Free the used memory.
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delete fec;
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return 0;
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}
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void quad_trans(double u, double v, double &x, double &y, bool log = false)
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{
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double a = a_; // Radius of disc
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double d = 4.0 * a * (M_SQRT2 - 2.0 * a) * (1.0 - 2.0 * v);
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double v0 = (1.0 + M_SQRT2) * (M_SQRT2 * a - 2.0 * v) *
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((4.0 - 3 * M_SQRT2) * a +
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(8.0 * (M_SQRT2 - 1.0) * a - 2.0) * v) / d;
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double r = 2.0 * ((M_SQRT2 - 1.0) * a * a * (1.0 - 4.0 *v) +
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2.0 * (1.0 + M_SQRT2 *
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(1.0 + 2.0 * (2.0 * a - M_SQRT2 - 1.0) * a)) * v * v
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) / d;
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double t = asin(v / r) * u / v;
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if (log)
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{
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mfem::out << "u, v, r, v0, t "
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<< u << " " << v << " " << r << " " << v0 << " " << t
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<< endl;
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}
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x = r * sin(t);
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y = r * cos(t) - v0;
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}
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void trans(const Vector &u, Vector &x)
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{
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double tol = 1e-4;
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if (u[1] > 0.5 - tol || u[1] < -0.5 + tol)
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{
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x = u;
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return;
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}
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if (u[0] > 1.0 - tol || u[0] < -1.0 + tol || fabs(u[0]) < tol)
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{
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x = u;
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return;
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}
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if (u[0] > 0.0)
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{
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if (u[1] > fabs(u[0] - 0.5))
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{
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quad_trans(u[0] - 0.5, u[1], x[0], x[1]);
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x[0] += 0.5;
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return;
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}
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if (u[1] < -fabs(u[0] - 0.5))
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{
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quad_trans(u[0] - 0.5, -u[1], x[0], x[1]);
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x[0] += 0.5;
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x[1] *= -1.0;
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return;
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}
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if (u[0] - 0.5 > fabs(u[1]))
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{
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quad_trans(u[1], u[0] - 0.5, x[1], x[0]);
|
|
x[0] += 0.5;
|
|
return;
|
|
}
|
|
if (u[0] - 0.5 < -fabs(u[1]))
|
|
{
|
|
quad_trans(u[1], 0.5 - u[0], x[1], x[0]);
|
|
x[0] *= -1.0;
|
|
x[0] += 0.5;
|
|
return;
|
|
}
|
|
}
|
|
else
|
|
{
|
|
if (u[1] > fabs(u[0] + 0.5))
|
|
{
|
|
quad_trans(u[0] + 0.5, u[1], x[0], x[1]);
|
|
x[0] -= 0.5;
|
|
return;
|
|
}
|
|
if (u[1] < -fabs(u[0] + 0.5))
|
|
{
|
|
quad_trans(u[0] + 0.5, -u[1], x[0], x[1]);
|
|
x[0] -= 0.5;
|
|
x[1] *= -1.0;
|
|
return;
|
|
}
|
|
if (u[0] + 0.5 > fabs(u[1]))
|
|
{
|
|
quad_trans(u[1], u[0] + 0.5, x[1], x[0]);
|
|
x[0] -= 0.5;
|
|
return;
|
|
}
|
|
if (u[0] + 0.5 < -fabs(u[1]))
|
|
{
|
|
quad_trans(u[1], -0.5 - u[0], x[1], x[0]);
|
|
x[0] *= -1.0;
|
|
x[0] -= 0.5;
|
|
return;
|
|
}
|
|
}
|
|
x = u;
|
|
}
|
|
|
|
Mesh * GenerateSerialMesh(int ref)
|
|
{
|
|
Mesh * mesh = new Mesh(2, 29, 16, 24, 2);
|
|
|
|
int vi[4];
|
|
|
|
for (int i=0; i<2; i++)
|
|
{
|
|
int o = 13 * i;
|
|
vi[0] = o + 0; vi[1] = o + 3; vi[2] = o + 4; vi[3] = o + 1;
|
|
mesh->AddQuad(vi);
|
|
|
|
vi[0] = o + 1; vi[1] = o + 4; vi[2] = o + 5; vi[3] = o + 2;
|
|
mesh->AddQuad(vi);
|
|
|
|
vi[0] = o + 5; vi[1] = o + 8; vi[2] = o + 9; vi[3] = o + 2;
|
|
mesh->AddQuad(vi);
|
|
|
|
vi[0] = o + 8; vi[1] = o + 12; vi[2] = o + 15; vi[3] = o + 9;
|
|
mesh->AddQuad(vi);
|
|
|
|
vi[0] = o + 11; vi[1] = o + 14; vi[2] = o + 15; vi[3] = o + 12;
|
|
mesh->AddQuad(vi);
|
|
|
|
vi[0] = o + 10; vi[1] = o + 13; vi[2] = o + 14; vi[3] = o + 11;
|
|
mesh->AddQuad(vi);
|
|
|
|
vi[0] = o + 6; vi[1] = o + 13; vi[2] = o + 10; vi[3] = o + 7;
|
|
mesh->AddQuad(vi);
|
|
|
|
vi[0] = o + 0; vi[1] = o + 6; vi[2] = o + 7; vi[3] = o + 3;
|
|
mesh->AddQuad(vi);
|
|
}
|
|
|
|
vi[0] = 0; vi[1] = 6; mesh->AddBdrSegment(vi, 1);
|
|
vi[0] = 6; vi[1] = 13; mesh->AddBdrSegment(vi, 1);
|
|
vi[0] = 13; vi[1] = 19; mesh->AddBdrSegment(vi, 1);
|
|
vi[0] = 19; vi[1] = 26; mesh->AddBdrSegment(vi, 1);
|
|
|
|
vi[0] = 28; vi[1] = 22; mesh->AddBdrSegment(vi, 2);
|
|
vi[0] = 22; vi[1] = 15; mesh->AddBdrSegment(vi, 2);
|
|
vi[0] = 15; vi[1] = 9; mesh->AddBdrSegment(vi, 2);
|
|
vi[0] = 9; vi[1] = 2; mesh->AddBdrSegment(vi, 2);
|
|
|
|
for (int i=0; i<2; i++)
|
|
{
|
|
int o = 13 * i;
|
|
vi[0] = o + 3; vi[1] = o + 7; mesh->AddBdrSegment(vi, 3 + i);
|
|
vi[0] = o + 7; vi[1] = o + 10; mesh->AddBdrSegment(vi, 3 + i);
|
|
vi[0] = o + 10; vi[1] = o + 11; mesh->AddBdrSegment(vi, 3 + i);
|
|
vi[0] = o + 11; vi[1] = o + 12; mesh->AddBdrSegment(vi, 3 + i);
|
|
vi[0] = o + 12; vi[1] = o + 8; mesh->AddBdrSegment(vi, 3 + i);
|
|
vi[0] = o + 8; vi[1] = o + 5; mesh->AddBdrSegment(vi, 3 + i);
|
|
vi[0] = o + 5; vi[1] = o + 4; mesh->AddBdrSegment(vi, 3 + i);
|
|
vi[0] = o + 4; vi[1] = o + 3; mesh->AddBdrSegment(vi, 3 + i);
|
|
}
|
|
|
|
double d[2];
|
|
double a = a_ / M_SQRT2;
|
|
|
|
d[0] = -1.0; d[1] = -0.5; mesh->AddVertex(d);
|
|
d[0] = -1.0; d[1] = 0.0; mesh->AddVertex(d);
|
|
d[0] = -1.0; d[1] = 0.5; mesh->AddVertex(d);
|
|
|
|
d[0] = -0.5 - a; d[1] = -a; mesh->AddVertex(d);
|
|
d[0] = -0.5 - a; d[1] = 0.0; mesh->AddVertex(d);
|
|
d[0] = -0.5 - a; d[1] = a; mesh->AddVertex(d);
|
|
|
|
d[0] = -0.5; d[1] = -0.5; mesh->AddVertex(d);
|
|
d[0] = -0.5; d[1] = -a; mesh->AddVertex(d);
|
|
d[0] = -0.5; d[1] = a; mesh->AddVertex(d);
|
|
d[0] = -0.5; d[1] = 0.5; mesh->AddVertex(d);
|
|
|
|
d[0] = -0.5 + a; d[1] = -a; mesh->AddVertex(d);
|
|
d[0] = -0.5 + a; d[1] = 0.0; mesh->AddVertex(d);
|
|
d[0] = -0.5 + a; d[1] = a; mesh->AddVertex(d);
|
|
|
|
d[0] = 0.0; d[1] = -0.5; mesh->AddVertex(d);
|
|
d[0] = 0.0; d[1] = 0.0; mesh->AddVertex(d);
|
|
d[0] = 0.0; d[1] = 0.5; mesh->AddVertex(d);
|
|
|
|
d[0] = 0.5 - a; d[1] = -a; mesh->AddVertex(d);
|
|
d[0] = 0.5 - a; d[1] = 0.0; mesh->AddVertex(d);
|
|
d[0] = 0.5 - a; d[1] = a; mesh->AddVertex(d);
|
|
|
|
d[0] = 0.5; d[1] = -0.5; mesh->AddVertex(d);
|
|
d[0] = 0.5; d[1] = -a; mesh->AddVertex(d);
|
|
d[0] = 0.5; d[1] = a; mesh->AddVertex(d);
|
|
d[0] = 0.5; d[1] = 0.5; mesh->AddVertex(d);
|
|
|
|
d[0] = 0.5 + a; d[1] = -a; mesh->AddVertex(d);
|
|
d[0] = 0.5 + a; d[1] = 0.0; mesh->AddVertex(d);
|
|
d[0] = 0.5 + a; d[1] = a; mesh->AddVertex(d);
|
|
|
|
d[0] = 1.0; d[1] = -0.5; mesh->AddVertex(d);
|
|
d[0] = 1.0; d[1] = 0.0; mesh->AddVertex(d);
|
|
d[0] = 1.0; d[1] = 0.5; mesh->AddVertex(d);
|
|
|
|
mesh->FinalizeTopology();
|
|
|
|
mesh->SetCurvature(1, true);
|
|
|
|
// Stitch the ends of the stack together
|
|
{
|
|
Array<int> v2v(mesh->GetNV());
|
|
for (int i = 0; i < v2v.Size() - 3; i++)
|
|
{
|
|
v2v[i] = i;
|
|
}
|
|
// identify vertices on the narrow ends of the rectangle
|
|
v2v[v2v.Size() - 3] = 0;
|
|
v2v[v2v.Size() - 2] = 1;
|
|
v2v[v2v.Size() - 1] = 2;
|
|
|
|
// renumber elements
|
|
for (int i = 0; i < mesh->GetNE(); i++)
|
|
{
|
|
Element *el = mesh->GetElement(i);
|
|
int *v = el->GetVertices();
|
|
int nv = el->GetNVertices();
|
|
for (int j = 0; j < nv; j++)
|
|
{
|
|
v[j] = v2v[v[j]];
|
|
}
|
|
}
|
|
// renumber boundary elements
|
|
for (int i = 0; i < mesh->GetNBE(); i++)
|
|
{
|
|
Element *el = mesh->GetBdrElement(i);
|
|
int *v = el->GetVertices();
|
|
int nv = el->GetNVertices();
|
|
for (int j = 0; j < nv; j++)
|
|
{
|
|
v[j] = v2v[v[j]];
|
|
}
|
|
}
|
|
mesh->RemoveUnusedVertices();
|
|
mesh->RemoveInternalBoundaries();
|
|
}
|
|
mesh->SetCurvature(3, true);
|
|
|
|
for (int l = 0; l < ref; l++)
|
|
{
|
|
mesh->UniformRefinement();
|
|
}
|
|
|
|
mesh->Transform(trans);
|
|
|
|
return mesh;
|
|
}
|
|
|
|
double IntegrateBC(GridFunction & x, Array<int> &bdr, bool h1)
|
|
{
|
|
FiniteElementSpace * fespace = x.FESpace();
|
|
ConstantCoefficient one(1.0);
|
|
|
|
// Integrate the basis functions along the given boundary
|
|
LinearForm lf(fespace);
|
|
if (h1)
|
|
{
|
|
lf.AddBoundaryIntegrator(new BoundaryLFIntegrator(one), bdr);
|
|
}
|
|
else
|
|
{
|
|
lf.AddBdrFaceIntegrator(new BoundaryLFIntegrator(one), bdr);
|
|
}
|
|
lf.Assemble();
|
|
|
|
// Compute the integral of x along the given boundary
|
|
return lf(x);
|
|
}
|
|
|
|
double IntegrateNGradBC(GridFunction &u, VectorCoefficient &nCoef,
|
|
Array<int> &bdr, bool h1,
|
|
OperatorHandle &M, double a)
|
|
{
|
|
FiniteElementSpace * fespace = u.FESpace();
|
|
|
|
// Compute the operator n.Grad
|
|
BilinearForm nd(fespace);
|
|
nd.AddDomainIntegrator(new MixedDirectionalDerivativeIntegrator(nCoef));
|
|
nd.Assemble();
|
|
|
|
// Apply the operator to the solution vector
|
|
LinearForm b(fespace);
|
|
nd.Mult(u, b);
|
|
|
|
// Solve for n.Grad(u)
|
|
Vector X;
|
|
X.SetSize(fespace->GetTrueVSize());
|
|
X = 0.0;
|
|
|
|
CGSolver mcg;
|
|
mcg.SetRelTol(1e-12);
|
|
mcg.SetMaxIter(2000);
|
|
mcg.SetPrintLevel(0);
|
|
mcg.SetOperator(*M);
|
|
mcg.Mult(b, X);
|
|
|
|
GridFunction ndu(fespace);
|
|
ndu.SetFromTrueDofs(X);
|
|
|
|
// For Robin BCs we add a * u
|
|
if (a != 0.0)
|
|
{
|
|
ndu.Add(a, u);
|
|
}
|
|
|
|
// Integrate n.Grad(u) + a * u along the given boundary
|
|
return IntegrateBC(ndu, bdr, h1);
|
|
}
|